A physically based model for calculating contributing area on

hillslopes and along valley bottoms

John B. Lindsay

Department of Geography, University of Western Ontario, London, Ontario, Canada

Received 9 August 2003; accepted 12 September 2003; published 3 December 2003.

[1] Most existing methods of calculating contributing area are unable to accuratelymodel the pattern of contributing area on hillslopes and along valley bottoms. This paperdescribes a new flow algorithm, the adjustable dispersion routing algorithm (ADRA).Rather than calculating contributing area using predetermined flow characteristics that areinsensitive to location in the landscape, ADRA predicts channel location and adjustsflow characteristics accordingly. ADRA increases the degree of flow divergence downslopefrom divides until a channel head is detected. Channel head locations are estimated on thebasis of a user-defined threshold of an area-slope function. Therefore the algorithmovercomes the problems of aggregated flow on hillslopes and divergent flow along valleybottoms. The pattern of catchment area produced by ADRAwas compared with similarpatterns calculated using a similar flow algorithm for a variety of natural landscapes. ADRAproduced patterns of contributing area that were more consistent with the theory of channelinitiation. INDEX TERMS: 1824 Hydrology: Geomorphology (1625); 1848 Hydrology: Networks;1860 Hydrology: Runoff and streamflow; 1894 Hydrology: Instruments and techniques; KEYWORDS: flow

routing, contributing area, digital elevation models, stream network extraction, channel initiation

Citation: Lindsay, J. B., A physically based model for calculating contributing area on hillslopes and along valley bottoms,

Water Resour. Res., 39(12), 1332, doi:10.1029/2003WR002576, 2003.

1. Introduction

[2] Contributing area is the area upslope of a location in acatchment from which runoff is captured. In practice, con-tributing area is calculated per unit contour length, referred toas the specific catchment area, a. The utility of a as a surrogatefor runoff volume makes it an essential parameter formodeling hydrologic, geomorphic, and other environmentalprocesses. Specific catchment area has been used to estimatesoil wetness [Beven and Kirkby, 1979], soil erosion anddeposition potential [Mitasova et al., 1996], and to extractstream networks and watersheds [Jenson and Domingue,1988] from digital elevation models (DEMs). Althoughdigital terrain data are now abundant and software packagesfor analyzing these data are common, a remains inherentlydifficult to estimate accurately [Gallant et al., 2000].[3] Specific catchment area is calculated using flow

routing algorithms that direct and accumulate runoff overDEMs. Flow routing algorithms differ in the way that theycalculate flow direction and in the method used to divideflow between each downslope neighbor. Some authors havecategorized flow algorithms based on whether they allowfor divergence [e.g., Wolock and McCabe, 1995]. Hencealgorithms that are unable to disperse flow are single-flow-direction (SFD) algorithms, and those that are capable ofdivergence are multiple-flow-direction (MFD) algorithms.The most common SFD algorithm is referred to as steepestdescent or D8 [OCallaghan and Mark, 1984]. Commonlyused MFD algorithms include FD8 [Freeman, 1991; Quinnet al., 1991], DEMON [Costa-Cabral and Burges, 1994],and D1 [Tarboton, 1997].

[4] Most existing flow algorithms do not explicitly differ-entiate between catchment hillslopes and channels, which isan important distinction given the contrast in geomorphic andhydrological processes operating on both. SFD algorithmsare suited to modeling incisive channelized flow along valleybottoms but are unable to simulate divergence on hillslopes.MFD algorithms yield more realistic diffusive flow patternson divergent hillslopes, but often result in braiding alongvalley bottoms. Braiding is an undesirable artifact that cancause discontinuous stream networks [Gallant and Wilson,2000] and inappropriately decrease a downstream [Quinn etal., 1995]. The problem with existing flow algorithms is thatthe amount of divergence is determined by local slope orcurvature without consideration of whether divergent flow isappropriate given the relative position of each cell in thelandscape. The challenge in overcoming this problem is thatthe flow algorithm must estimate channel network extentduring processing. This paper describes a flow routingscheme that has been developed to address these issues.

2. Background

[5] The FD8 algorithm is unique among existing flowrouting algorithms in that it is possible to adjust the overalldegree of divergence. The Freeman [1991] implementationof FD8 calculates the fraction (F ) of a apportioned to eachdownslope neighbor, i, by,

Fi max0; tan Spi P8

j1max0; tan Spj

1

where S is the slope between cells. Summing Fiak for eachupslope neighbor k yields ai. A larger value of p results in a

Copyright 2003 by the American Geophysical Union.0043-1397/03/2003WR002576$09.00

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WATER RESOURCES RESEARCH, VOL. 39, NO. 12, 1332, doi:10.1029/2003WR002576, 2003

greater fraction of a being apportioned to the lowestneighbor. As p ! 1, F ! 1 for the neighbor of steepestdescent and F ! 0 for all other neighbors (Figure 1).[6] Many implementations of FD8 use a value for p

between 1.16 but this practice results in a compromisein terms of modeling flow divergence. That is, single pvalues attempt to fit a catchment-wide average of flowdivergence that results in flow on hillslopes appearing tooconvergent and braiding along valley bottoms. The sameproblem of hard-wired flow properties, which are indifferentto a cells location relative to the stream network, exists forthe DEMON and D1 algorithms. Some researchers havesuggested using hybrid flow algorithms to compensatefor these problems. For example, equation (1) can beused on hillslopes until convergent topography is detected[Freeman, 1991] or a channel initiation threshold (CIT)catchment area is reached [Quinn et al., 1991], beyondwhich a is calculated using the D8 SFD flow algorithm. Onecriticism with this approach is that irregularities, or abruptchanges, occur at the transition to channelized flow [Gallantand Wilson, 2000]. Quinn et al. [1995] overcame thisproblem by increasing p continuously downslope until theCIT was reached, creating a downslope feedback betweenthe degree of flow divergence and a. An adjustable powerterm determined how rapidly flow became convergent. Thispattern of increasingly convergent flow toward channelheads is consistent with field observations [Quinn et al.,1995].[7] There are several problems, however, with using p to

adjust the degree of divergence. First, F is a nonlinearfunction of p, and therefore changing p an equivalentamount results in an unequal increase in the degree of flowconvergence (Figure 1). Furthermore, when the slopes toneighboring cells are similar, substantial partitioning canoccur even when p is large. When two or more cellshave equal slopes, equation (1) will partition flow evenif p equals infinity (Figure 1). This problem is furthercompounded by the practical limitation that large exponentscan cause overflow errors in most computers.

[8] In addition to these technical issues, the threshold aused in the hybrid and Quinn et al. [1995] implementationsof FD8 may not be physically realistic. Montgomery andFoufoula-Georgiou [1993] concluded that models withconstant critical support areas for channel maintenanceare theoretically and empirically less appropriate thanslope-dependent critical support areas. This is becausesmaller contributing areas are needed to initiate channelson steeper slopes. If overland flow is assumed to be themechanism for channel initiation, channel heads occur whenthe basal shear stress (tb) exceeds the critical shear stressof the ground surface (tcr). On the basis of a laminarflow model, Montgomery and Foufoula-Georgiou [1993]showed that the critical specific catchment area (acr)required for tb > tcr is,

acr Ctan q2 C / t3cr; q

1r 2

where q is the local slope. C is a constant with unitsof length, which is proportional to tcr

3 and inverselyproportional to the steady state rainfall intensity, qr .[9] On the basis of equation (2), the transition from

unchannelized to channelized flow occurs where

atan q2 C 3

Montgomery and Dietrich [1992] and Montgomery andFoufoula-Georgiou [1993] found that this model couldpredict channel head locations. Therefore equation (3) is anempirical method for determining the channel networkextent, although scatter in the relation can result from thespatial variation in tcr [Istanbulluoglu et al., 2002] and theassumption that the channel network is in a state of long-term equilibrium. Equation (3) is only necessarily valid atchannel heads, i.e., a(tanq)2 can decrease downslope.[10] On the basis of field observations, Montgomery and

Dietrich [1992] reported values of C between 25 m and

Figure 1. Fraction of a received by two downslope neighbors of varying slopes using the Freeman[1991] concentration parameter p.

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200 m for the Tennessee Valley area of northern California.Dietrich et al. [1993] and Montgomery and Foufoula-Georgiou [1993] present several methods for estimatingC for different regions. One of the most practical of theseestimation methods is based on evidence that channelhead locations define the limit to convergent topography[Montgomery and Dietrich, 1992]. Montgomery andFoufoula-Georgiou [1993] suggest that an appropriate valueof C reflects the smallest value that does not result in theextension of the channel network onto the surroundingplanar or divergent topography. Thus C is the smallestvalue that does not result in feathering of the channelnetwork along headwater channels.

3. Adjustable Dispersion Routing Algorithm(ADRA)

[11] The adjustable dispersion routing algorithm (ADRA)was developed to address some of the problems withexisting algorithms. ADRA calculates Fi as,

Fi max0; tan SiP8j1

max0; tan Sj n ki max0; tan SiP8

j1max0; tan Sj

0BBB@

1CCCA 4

where the concentration parameter, u, ranges from zero toone and k equals one for the first cell detected with themaximum downward slope and zero for all other cells. Forthe cell with the maximum downward slope, F(u) is a linefrom S/S to one, and for all other cells with downwardslopes, F(u) is a line from S/S to zero (Figure 2).[12] ADRA has been designed to vary u with the area-

slope function a(tanq)2. If a cell has a low a(tanq)2 value(i.e., a(tanq)2 < C) then it is above the channel head and alow u should be used to divide a between downslopeneighbors. Similarly, a high a(tanq)2 value indicates chan-nelized flow and u should equal one. Once C is encoun-

tered, flow divergence is undesirable and all subsequentdownslope cells are treated as channel cells. Thus

n t atan q2 < C

1 atan q2 C

5

where the transition function, t, is a curve that risesmonotonically upward such that t(0) = 0 and t(C) = 1.Although other models may be suitable (e.g., a sigmoidalcurve), a linear transition function was used in this paper, ofthe form:

t atan q2

C; atan q2 < C 6

[13] Using the recursive drainage accumulation approach[Mark, 1988; Freeman, 1991], a is calculated for a cell onlyafter each upslope neighbor has been solved. Thus, for anycell i, a(tanq)2 and u are known for each upslope neighbor kand ai = Fiak.

4. Flow Algorithm Comparison

[14] ADRA was compared with the Quinn et al. [1995]variable-p implementation of FD8, because this algorithmproduces the most logically consistent pattern of a (i.e.,divergent flow on hillslopes and convergent flow invalleys). Several researchers have used artificial surfaces(e.g., planes and cones) to evaluate flow algorithms becausethe actual contributing area is known for all locations [e.g.,Freeman, 1991; Costa-Cabral and Burges, 1993; Tarboton,1997]. However, this approach was inappropriate for eval-uating ADRA and the Quinn et al. [1995] algorithmbecause artificial surfaces lack stream channels. The onlyalternative therefore was to compare the visual patterns of agenerated by both flow algorithms in a variety of naturalsettings. Nevertheless, this approach offered valuableinsight into the applicability of ADRA because theoryindicates, at least qualitatively, what the spatial pattern ofa should look like.

Figure 2. Fraction of a received by two downslope neighbors of varying slopes using the ADRAconcentration parameter n.

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Figure 3. Patterns of catchment area calculated using ADRA and the Quinn et al. [1995] algorithm for(a) a mountainous catchment in coastal Washington, (b) a moderately steep catchment in northernVermont, (c) a low-relief catchment in Texas, and (d) a low-relief catchment in Oklahoma.

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[15] Four study catchments were identified in differentphysiographic regions. These catchments representedranges in relief (i.e., 56 to 786 m) and climate (i.e., marine,humid sub-tropical, humid continental, and midlatitudedry). Shuttle Radar Topography Mission (SRTM-1) DEMs[Farr and Kobrick, 2000] were used as the necessary inputto the flow algorithms. The original DEMs, with gridresolutions of 1 arc-second, were projected into UTMcoordinates and resampled to 30 m grids. The verticalprecision of SRTM-1 DEMs is 1 m.[16] ADRA and the Quinn et al. [1995] algorithm pro-

duced very similar patterns of a, with divergent flow onhillslopes and fully convergent flow along valley bottoms(Figure 3). Stream channels were evident in the flowaccumulation maps as single-cell wide networks of higha-values. Braiding did not occur using either flow algo-rithm. Stream network extent was marked by graduallyincreasing flow convergence near channel heads in eachstudy catchment. The differences between the flow accu-mulations maps produced by ADRA and the Quinn et al.[1995] algorithm were most pronounced near valley headswhere the rate of downslope flow convergence and thepositioning and number of channel heads differed. Thesedissimilarities were most apparent in Figure 3b because ofthe large hillslopes and low drainage density of the Vermontstudy catchment. Conversely, the differences between thevisual patterns of a produced by the two flow algorithmswere negligible for the low-relief catchment in Oklahoma(Figure 3d), because of the relatively high drainage densityand little variation in slope.[17] The relation between q and the estimated value of a

at channel heads is illustrated in Figure 4 for the mountain-ous catchment in Washington (Figure 3a). Figure 4 clearlydemonstrates that a varied inversely with slope for ADRAwhile the Quinn et al. [1995] algorithm did not exhibit asimilar relation. Thus the spatial pattern of a derived usingADRA was more consistent with the theory of channelinitiation. However, Figure 4 also demonstrates that ADRAcan be more susceptible to a particular type of DEM error.

The three data points in Figure 4 with slopes greater than60 resulted from missing data in the SRTM-1 data, whichoccur on mountainsides. These missing data (given thevalue of 32768) were filled during preprocessing, yieldingspurious cliff faces on the upslope side of the pit. Thesesites are recognized as channel heads by ADRA despitetheir small catchment areas because of their erroneouslyhigh slopes. The Quinn et al. [1995] algorithm does notinclude local slope in its estimation of channel extent andtherefore is not similarly susceptible to this type of error.ADRA would not identify channel heads in these areas ifmissing data were removed using an interpolation schemethat approximates the original hillslope rather than using pitfilling.

5. Summary

[18] Most existing SFD and MFD routing algorithms areinadequate for modeling contributing area over entirecatchments. Rather than basing the degree of divergenceon local slope alone, ADRA divides flow using local slopeand an estimate of the cells position relative to channelheads. Thus, by incorporating a model of channel initiationdirectly into the flow calculation, ADRA can modelcatchment area on hillslopes and along valley bottoms.The transition between hillslope and channel flow is basedon a physically realistic model, capable of locating channelheads. Also, the proportion of the catchment area that acell receives is a linear function of the concentrationparameter, which has a range from zero to one. In contrast,exponent based concentration parameters divide flow non-linearly and range from zero to infinity. A comparisonbetween ADRA and the Quinn et al. [1995] flow algo-rithm, across a range of basin types, showed that the twomethods produce visually similar spatial patterns of a, butdiffer in the representation of flow accumulation nearvalley heads. ADRAs flow accumulation patterns werefound to be more consistent with the theory of channelinitiation. However, if DEMs contain missing data values,ADRA can yield artifact channels because of erroneousslopes. Thus care is needed to correctly handle missingdata values.

[19] Acknowledgments. The author thanks I. Creed and three anon-ymous reviewers for their invaluable comments and advice. This work wassupported by the Ontario Graduate Scholarship.

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J. B. Lindsay, Department of Geography, University of Western Ontario,

London, Ontario, Canada N6A 5C2. ( jblindsa@uwo.ca)

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